This article addresses the boundary controllability problem for a class of third order in time PDE, known as Moore-Gibson-Thompson equation, with a control supported on the boundary. It is shown that it is not spectrally controllable, which means that nontrivial finite linear combination of eigenvectors can be driven to zero in finite time. This implies that the Moore-Gibson-Thompson equation is not exact and null controllable. However, the approximate controllability will be proved.